Three angles meet at a point. Two of the angles are 130 degrees and 145 degrees, and the third angle is x degrees.
(Total for Question 1 is 2 marks)
2
Two angle facts about intersecting and straight lines.
(a)Angle p and an angle of 64 degrees lie on a straight line, on the same side of a point. Work out the size of angle p.(1)
(b)Two straight lines cross. One of the angles formed is (3x + 10) degrees, and the angle vertically opposite it is (5x - 40) degrees. Work out the value of x, and hence the size of the angle (3x + 10) degrees.(3)
(Total for Question 2 is 4 marks)
3
Two parallel lines are cut by a transversal. One of the angles formed, angle a, is 72 degrees.
(a)Angle b is corresponding to angle a. Write down the size of angle b, giving a reason.(2)
(b)Angle c is co-interior (allied) with angle a, on the same side of the transversal. Work out the size of angle c, giving a reason.(2)
(c)Angle d is alternate to angle a. Write down the size of angle d, giving a reason.(1)
(Total for Question 3 is 5 marks)
4
Interior angles of polygons.
(a)Calculate the sum of the interior angles of a nonagon (a 9-sided polygon).(2)
(b)A hexagonal (6-sided) traffic island is regular. Calculate the size of each interior angle.(2)
(c)An irregular pentagon has four interior angles of 100 degrees, 110 degrees, 95 degrees and 130 degrees. Calculate the size of the fifth interior angle.(2)
(Total for Question 4 is 6 marks)
5
Exterior angles of polygons.
(a)A regular decagon (a 10-sided polygon) is used as the pattern for a patio. Work out the size of each exterior angle.(2)
(b)Hence work out the size of each interior angle of the regular decagon.(2)
(c)A different regular polygon has an exterior angle of 24 degrees. Work out the number of sides of the polygon.(2)
(Total for Question 5 is 6 marks)
6
A school's L-shaped wildlife garden is formed from a rectangle measuring 12 m by 9 m, with a rectangular corner measuring 5 m by 4 m removed from one corner.
(a)Calculate the perimeter of the L-shaped garden.(4)
(b)Calculate the area of the L-shaped garden.(3)
(Total for Question 6 is 7 marks)
7
Calculate the area of each shape.
(a)A triangular sail has a base of 14 cm and a perpendicular height of 9 cm.(2)
(b)A parallelogram-shaped tile has a base of 8.5 cm and a perpendicular height of 6 cm.(2)
(c)A trapezium-shaped window pane has parallel sides of 7 cm and 11 cm, and a perpendicular distance of 5 cm between them.(2)
(Total for Question 7 is 6 marks)
8
A circular pond has a diameter of 8.4 m. Take π = 3.14 throughout this question.
(a)Calculate the circumference of the pond, giving your answer to 1 decimal place.(2)
(b)Calculate the area of the pond, giving your answer to 1 decimal place.(3)
(Total for Question 8 is 5 marks)
9
A cuboid-shaped water tank has length 1.5 m, width 0.8 m and height 1.2 m.
(a)Calculate the volume of the tank.(2)
(b)Calculate the total surface area of the tank.(3)
(Total for Question 9 is 5 marks)
10
A prism has a cross-section that is a right-angled triangle with base 6 cm and height 8 cm. The length of the prism is 15 cm.
(a)Calculate the area of the triangular cross-section.(2)
(b)Calculate the volume of the prism.(2)
(Total for Question 10 is 4 marks)
11
A right-angled triangle has two shorter sides of length 9 cm and 12 cm.
(Total for Question 11 is 3 marks)
12
A right-angled triangle has a hypotenuse of 26 cm and one shorter side of 24 cm.
(Total for Question 12 is 3 marks)
13
A ladder of length 6.5 m leans against a vertical wall, with its foot on horizontal ground.
(a)The foot of the ladder is 2.5 m from the base of the wall. Calculate how high up the wall the ladder reaches.(3)
(b)The foot of the ladder is now moved so that it is 3.6 m from the base of the wall. Calculate how high up the wall the ladder now reaches, giving your answer to 2 decimal places.(3)
(Total for Question 13 is 6 marks)
14
Pythagoras' theorem and its converse.
(a)A rectangular field measures 40 m by 30 m. Calculate the length of the diagonal path across the field.(3)
(b)A triangular flowerbed has sides of length 7 m, 24 m and 25 m. Show that the flowerbed is right-angled, and state which angle is the right angle.(3)
(Total for Question 14 is 6 marks)
15
A is the point (2, 3) and B is the point (10, 9).
(a)Find the midpoint of AB.(2)
(b)Calculate the distance AB.(3)
(Total for Question 15 is 5 marks)
16
Triangle T has vertices A(1, 1), B(4, 1) and C(1, 3).
(a)Triangle T is reflected in the line x = 0 (the y-axis) to give triangle T'. Write down the coordinates of the image of vertex B.(2)
(b)Triangle T is instead rotated 90 degrees clockwise about the origin to give triangle T''. Write down the coordinates of the image of vertex C.(2)
(Total for Question 16 is 4 marks)
17
Shape S is drawn on a coordinate grid.
(a)Shape S is translated by the vector (5, -2). Point P(-3, 4) lies on S. Find the coordinates of the image of P.(2)
(b)Shape S is instead enlarged by scale factor 3, centre the origin. Point Q(2, -1) lies on S. Find the coordinates of the image of Q.(2)
(c)Shape S is instead enlarged by scale factor 1/2, centre (4, 2). Point R(6, 2) lies on S. Find the coordinates of the image of R.(3)
(Total for Question 17 is 7 marks)
18
A map of the area around the villages of Ashcombe and Coalworth has a scale of 1 : 25000.
(a)On the map, the distance between Ashcombe and Coalworth is 6.4 cm. Work out the real distance between the villages, in km.(3)
(b)A reservoir near Coalworth has a real area of 4.5 km2. Work out the area representing the reservoir on the map, in cm2.(4)
(Total for Question 18 is 7 marks)
19
Constructions using a ruler and compasses.
(a)Draw a straight line segment AB of length 8 cm. Using only a ruler and compasses, construct the perpendicular bisector of AB. Show all your construction arcs.(2)
(b)Draw an angle ABC = 70 degrees, with BA and BC each about 6 cm long. Using only a ruler and compasses, construct the bisector of angle ABC. Show all your construction arcs.(2)
(c)Using a ruler and protractor, construct triangle PQR with PQ = 9 cm, angle P = 50 degrees and angle Q = 65 degrees.(3)
(Total for Question 19 is 7 marks)
20
Freya cycles from her home to Lower Norbury, a distance of 45 km, taking 2.5 hours.
(a)Work out Freya's average speed for this part of the journey, in km/h.(2)
(b)Freya then cycles a further 30 km at an average speed of 24 km/h. Work out the time taken for this part of the journey, giving your answer in hours and minutes.(3)
(c)Work out Freya's average speed, in km/h, for the whole 75 km journey.(3)
(Total for Question 20 is 8 marks)
21
A metal cube has a side length of 4 cm and a mass of 500 g.
(a)Calculate the volume of the cube.(2)
(b)Calculate the density of the metal, in g/cm3, giving your answer to 1 decimal place.(3)
(Total for Question 21 is 5 marks)
22
A cylindrical water butt has a radius of 35 cm and a height of 1.2 m. Take π = 3.14 throughout this question.
(a)Calculate the volume of the water butt, in cm3, giving your answer to 3 significant figures.(4)
(b)Given that 1 litre = 1000 cm3, work out the capacity of the water butt in litres, to the nearest litre.(2)
(Total for Question 22 is 6 marks)
23
A square-based pyramid has a square base of side 10 cm. The apex is directly above the centre of the base, and the vertical height of the pyramid is 12 cm.
(a)Work out the slant height, l, from the apex to the midpoint of a base edge, using Pythagoras' theorem with the vertical height and half the base side.(3)
(b)Work out the area of one triangular face of the pyramid.(2)
(c)Work out the total surface area of the pyramid (the four triangular faces plus the square base).(3)
(Total for Question 23 is 8 marks)
Mark scheme · CE.M29 Geometry, Measures and Constructions Depth
Question 1
M1 360 - 130 - 145 oe
A1 85 cao
Answer: x = 85 degrees
Question 2
(a) B1 116 cao
(a) Answer: p = 116 degrees
(b) M1 3x + 10 = 5x - 40 oe formed, using vertically opposite angles are equal
(b) A1 x = 25 cao
(b) A1 angle = 85 degrees cao, ft from their x
(b) Answer: x = 25, angle = 85 degrees
Question 3
(a) B1 72 cao
(a) B1 reason: corresponding angles are equal (oe, e.g. F-angles)
(a) Answer: b = 72 degrees, since corresponding angles are equal
(b) M1 180 - 72 oe
(b) A1 108 cao, with reason co-interior (allied) angles sum to 180 degrees
(b) Answer: c = 108 degrees, since co-interior angles sum to 180 degrees
(c) B1 72 cao, with reason alternate angles are equal (oe, e.g. Z-angles)
(c) Answer: d = 72 degrees, since alternate angles are equal
Question 4
(a) M1 (9 - 2) x 180 oe
(a) A1 1260 degrees cao
(a) Answer: 1260 degrees
(b) M1 (6 - 2) x 180 = 720, then 720 / 6 oe
(b) A1 120 degrees cao
(b) Answer: 120 degrees
(c) M1 (5 - 2) x 180 = 540, then 540 - 100 - 110 - 95 - 130 oe
(c) A1 105 degrees cao
(c) Answer: 105 degrees
Question 5
(a) M1 360 / 10 oe
(a) A1 36 degrees cao
(a) Answer: 36 degrees
(b) M1 180 - 36 oe, ft from part a
(b) A1 144 degrees cao, ft
(b) Answer: 144 degrees
(c) M1 360 / 24 oe
(c) A1 15 (sides) cao
(c) Answer: 15 sides
Question 6
(a) M1 12 - 5 = 7 (m), the length of the unlabelled edge parallel to the 12 m side
(a) M1 9 - 4 = 5 (m), the length of the unlabelled edge parallel to the 9 m side
(a) M1 12 + 9 + 5 + 4 + 7 + 5, summing all six edges
(a) A1 42 m cao
(a) Answer: 42 m
(b) M1 12 x 9 = 108 (area of full rectangle)
(b) M1 5 x 4 = 20 (area removed)
(b) A1 88 m2 cao
(b) Answer: 88 m2
Question 7
(a) M1 0.5 x 14 x 9 oe
(a) A1 63 cm2 cao
(a) Answer: 63 cm2
(b) M1 8.5 x 6 oe
(b) A1 51 cm2 cao
(b) Answer: 51 cm2
(c) M1 0.5 x (7 + 11) x 5 oe
(c) A1 45 cm2 cao
(c) Answer: 45 cm2
Question 8
(a) M1 3.14 x 8.4 oe
(a) A1 26.4 m (awrt 26.4)
(a) Answer: 26.4 m
(b) M1 radius = 8.4 / 2 = 4.2 (m)
(b) M1 3.14 x 4.22 oe
(b) A1 55.4 m2 (awrt 55.4)
(b) Answer: 55.4 m2
Question 9
(a) M1 1.5 x 0.8 x 1.2 oe
(a) A1 1.44 m3 cao
(a) Answer: 1.44 m3
(b) M1 three face areas found: 1.5 x 0.8 = 1.2, 1.5 x 1.2 = 1.8, 0.8 x 1.2 = 0.96
(b) M1 2 x (1.2 + 1.8 + 0.96) oe
(b) A1 7.92 m2 cao
(b) Answer: 7.92 m2
Question 10
(a) M1 0.5 x 6 x 8 oe
(a) A1 24 cm2 cao
(a) Answer: 24 cm2
(b) M1 24 x 15 oe, ft from part a
(b) A1 360 cm3 cao
(b) Answer: 360 cm3
Question 11
M1 92 + 122 oe (= 225)
M1√225
A1 15 cm cao
Answer: 15 cm
Question 12
M1 262 - 242 oe (= 100)
M1√100
A1 10 cm cao
Answer: 10 cm
Question 13
(a) M1 6.52 - 2.52 oe (= 36)
(a) M1√36
(a) A1 6 m cao
(a) Answer: 6 m
(b) M1 6.52 - 3.62 oe (= 29.29)
(b) M1√29.29
(b) A1 5.41 m (awrt 5.41)
(b) Answer: 5.41 m
Question 14
(a) M1 402 + 302 oe (= 2500)
(a) M1√2500
(a) A1 50 m cao
(a) Answer: 50 m
(b) M1 72 + 242 = 49 + 576 = 625 calculated
(b) A1 252 = 625, so 72 + 242 = 252, confirming (by the converse of Pythagoras' theorem) that the triangle is right-angled
(b) B1 the right angle is between the sides of 7 m and 24 m (i.e. opposite the 25 m side)
(b) Answer: The triangle is right-angled; the right angle is between the 7 m and 24 m sides.
(a) M1 reflecting in x = 0 negates the x-coordinate, y-coordinate unchanged
(a) A1 (-4, 1) cao
(a) Answer: (-4, 1)
(b) M1 rule (x, y) -> (y, -x) applied for a 90 degree clockwise rotation about the origin
(b) A1 (3, -1) cao
(b) Answer: (3, -1)
Question 17
(a) M1 (-3 + 5, 4 - 2) oe
(a) A1 (2, 2) cao
(a) Answer: (2, 2)
(b) M1 (2 x 3, -1 x 3) oe
(b) A1 (6, -3) cao
(b) Answer: (6, -3)
(c) M1 vector from centre to R found: (6 - 4, 2 - 2) = (2, 0)
(c) M1 scaled vector (2 x 1/2, 0 x 1/2) = (1, 0) added to the centre (4, 2)
(c) A1 (5, 2) cao
(c) Answer: (5, 2)
Question 18
(a) M1 6.4 x 25000 = 160000 (cm)
(a) M1 160000 / 100000 oe, converting cm to km
(a) A1 1.6 km cao
(a) Answer: 1.6 km
(b) M1 1 km = 100000 cm real; 100000 / 25000 = 4 (cm on map per km real), the linear map-to-real conversion
(b) M1 area scale factor = 42 = 16 (cm2 on map per km2 real)
(b) M1 4.5 x 16 oe
(b) A1 72 cm2 cao
(b) Answer: 72 cm2
Question 19
(a) C1 two pairs of arcs of equal radius (radius greater than 4 cm), one pair centred on A and one pair centred on B
(a) A1 correct straight line drawn through both points of intersection, within 2 mm and 2 degrees of the true perpendicular bisector
(a) Answer: The perpendicular bisector of AB, passing through its midpoint, 4 cm from both A and B.
(b) C1 one arc, centred at B, of any radius, crossing both arms BA and BC
(b) A1 bisector ray drawn from B through the intersection of two further equal-radius arcs, splitting angle ABC into two angles each within 2 degrees of 35 degrees
(b) Answer: The bisector splits angle ABC into two 35 degree angles.
(c) B1 PQ = 9 cm drawn accurately, within 2 mm
(c) B1 angle of 50 degrees drawn at P, within 2 degrees
(c) B1 angle of 65 degrees drawn at Q, within 2 degrees, with the two rays extended to meet accurately at R
(c) Answer: Triangle PQR with PQ = 9 cm, angle P = 50 degrees, angle Q = 65 degrees (and angle R = 65 degrees).
Question 20
(a) M1 45 / 2.5 oe
(a) A1 18 km/h cao
(a) Answer: 18 km/h
(b) M1 30 / 24 oe (= 1.25 hours)
(b) M1 0.25 hours = 15 minutes
(b) A1 1 hour 15 minutes cao
(b) Answer: 1 hour 15 minutes
(c) M1 total distance = 45 + 30 = 75 (km)
(c) M1 total time = 2.5 + 1.25 = 3.75 (hours), ft from part b
(c) A1 20 km/h cao
(c) Answer: 20 km/h
Question 21
(a) M1 43 oe
(a) A1 64 cm3 cao
(a) Answer: 64 cm3
(b) M1 density = mass / volume oe, formula seen or implied
(b) M1 500 / 64 (ft from part a)
(b) A1 7.8 g/cm3 (awrt 7.8)
(b) Answer: 7.8 g/cm3
Question 22
(a) M1 height converted to 120 cm (consistent units)
(a) M1 352 = 1225 found
(a) M1 3.14 x 1225 x 120 oe
(a) A1 462000 cm3 (awrt 462000, i.e. 4.62 x 105)
(a) Answer: 462000 cm3 (3 sf)
(b) M1 461580 / 1000 oe, ft from part a (using the unrounded volume)